Modelling the Non-Linear Dynamic Behaviour of a Laminated Composite Beam Made of Shape Memory Alloys Using the Multiple Time Scale Method

In this work, the nonlinear dynamic behavior of a shape memory alloy (SMA) beam, stratified in axial motion and subjected to transverse harmonic excitation, is studied. The polynomial constitutive model of Falk based on Landau’s theory is used to describe the thermomechanical characteristics of the shape memory alloy in the beam. The behavior of the substrate is described by a linear elastic law. To account for the effects of axial motion, the dynamic evolution equation of the SMA laminated beam is established from the physical equation, the force equilibrium conditions, the strain compatibility equation, and the constitutive relation, using the SMA polynomial function combined with the Galerkin procedure. The modulation equations near primary resonance are obtained using the second-order multiple time scales method. The effect of external and internal parameters on the responses to transverse vibrations is discussed. The numerical results obtained show that the SMA-reinforced composite structure can be adapted to the environmental conditions by adjusting the phase transition temperature and the thickness of the SMA layer.